## 11 February 2008

If the IRS had discovered the quadratic formula -- the algorithm for solving a quadratic equation, in the style of an IRS tax form. (This has been floating around for a while; I'm just reviving it.)

It seems silly here because everyone knows the quadratic formula. But the cubic formula is quite ugly, and takes up a couple lines; it's better expressed as an algorithm, which basically says to make a few changes of variables that give a quadratic and then solve the resulting quadratic. (Certainly if one were in the business of solving cubics, one would probably memorize the algorithm, not the formula.) And I don't even want to think about writing down an explicit formula for the general solution of the quartic in terms of radicals.

But tax forms are really just walking ordinary people through dealing with calculations involving certain equations. Sometimes I think taxation would make more sense to me if I actually saw the equations.

William said...

If I remember correctly the actual equation for tax isn't hard once you have the income variable X--it's merely a piecewise defined linear function. It's finding X that's the problem. As for the quartic...

http://planetmath.org/encyclopedia/QuarticFormula.html

The Galois-theoretic derivation is much easier to understand. Anonymous said...

The square root form could have been made even bigger by using a numerically more robust formula. In the standard formula (-B±√(B^2-4AC))/2A, compute one root x_1 by chosing the sign to match that of -B, then compute the root as x_2=C/Ax_1. This avoids the cancellation effects that could otherwise ruin the accuracy of the smaller root. Anonymous said...

I guess the moral of it is that there is little value in memorizing a formula (even for a quadratic). Learning how to derive a formula is much more valuable and intellectually stimulating.